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General Four-Terminal Network Analysis

Author: Leonard Krugman

The general four-terminal active network is fully described by the relationship between the input and output currents and voltages. Referring to Fig. 3-1, the general voltage (loop) equations are:

E1 = Z11I1 + Z12I2

and

E2 = Z21I1 + Z22I2

where

Z11 is the input impedance with the output open.
Z11 = E1/I1, when I2 = 0.
Z12 is the feedback or reverse transfer impedance with the input open.
Z12 = E1/I2, when I1 = 0.
Z21 is the forward transfer impedance with the output open.
Z21 = E2/I1, when I2 = 0.
Z22 is the output impedance with the input open.
Z22 = E2/I2, when I1 = 0.

The equivalent current (nodal) equations are

I1 = Y11E1 + Y12E2

and

I2 = Y21E1 + Y22E2

where

Y11 is the input admittance with the output shorted.
Y11 = I1/E1, where E2 = 0.
Y12 is the feedback or reverse transfer admittance with the input shorted.
Y12 = I1/E2, when E1 = O.
Y21 is the forward transfer admittance with the output shorted.
Y21 = I2/El, when E2 = O.
Y22 is the output admittance with the input shorted.
Y22 = I2/E2, when E1 = 0.

Amplification factors are the best general index of an active network. Since the general case may have amplification in both directions, definitions are included for forward and reverse directions.

The forward current amplification factor, α21, is equal to the negative ratio of the current at the shorted output terminals to the current at the input terminals.

a21 = -I2/I1 when E2 = 0

Then 0 = E2 = Z21I1 + Z22I2.

Solving these equations a21 = -I2/I1 = Z21/Z22 and in terms of admittance

a21 = -Y21/Y11

The reverse current amplification factor, α12, is equal to the negative ratio of the current at the shorted input terminals to the current at the output terrminals:

a12 = -I1/I2 when E1 = 0

Then 0 = E1 = Z11I1 + Z12I2.

Solving as before, a12 = -I1/I2 = Z12/Z11, and in terms of admittances

a12 = -Y12/Y22

The forward voltage amplification factory, μ21, is equal to the ratio of the open circuit output voltage to the input voltage. , when I2 = 0. On this basis, E1 = Z11I1 and E2 = Z21I1.

Thus and on an admittance basis

The reverse voltage amplification factor μ12 is equal to the ratio of the open circuit input voltage to the output voltage. when I1 = 0. Then E1 = Z12I2 and E2 = Z22I2. Thus In terms of admittance .

Last Update: 2010-11-17